Haut PDF Lambda-Calculus, Multiplicities and the pi-Calculus

Lambda-Calculus, Multiplicities and the pi-Calculus

Lambda-Calculus, Multiplicities and the pi-Calculus

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On differential interaction nets and the pi-calculus

On differential interaction nets and the pi-calculus

But one knows since the introduction of powerdomains by Plotkin in [Plo76] that denotational seman- tics can be extended with a reasonable amount of non-determinism, corresponding for instance to a non- deterministic choice operator – non-deterministic extensions of the lambda-calculus and of PCF have been designed, with this kind of operational features, and powerdomain-based denotational semantics. Even more drastically, if one renounces to the domain-theoretic viewpoint on semantics, or more precisely, to the fact that the domain interpreting the types should have some kind of built-in coherence, or compatibility notion, then there are no obstacles to define models of lambda-calculi, or of linear logic, which admit non-determinism under the guise of the possibility of defining arbitrary joins (or unions, or sums) of points.
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Algebras and Coalgebras in the Light Affine Lambda Calculus

Algebras and Coalgebras in the Light Affine Lambda Calculus

The syntax of the Light Affine Lambda Calculus (LALC) is in- spired by the restrictions provided by Light Affine Logic. The fo- cus of our work is on the expressivity of the calculus rather than on other properties, so to make our examples more clear we adopt an explicitly typed version of LALC. The types and the terms of LALC are presented in Figure 1 . As basic type we consider only the multiplicative unit 1, while as type constructors we consider the linear implication (, the tensor product ⊗, and the additive dis- junction ⊕. Moreover, we have type variables X and type universal and existential quantifications: ∀X.τ , and ∃X.τ . We also have two modalities !, and §. The connectives ⊗, ⊕ and the existential quan- tification ∃X.τ can be defined by using only the linear implication (, the modalities !, §, and the universal quantification ∀ but we prefer here to consider them as primitive. The reason behind this choice is that in the second part of the paper we will introduce ex- plicit rules for distributing the § modality over ⊗ and ⊕, so it is natural to consider them as primitive.
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Typechecking in the lambda-Pi-Calculus Modulo : Theory and Practice

Typechecking in the lambda-Pi-Calculus Modulo : Theory and Practice

Ces idées ont été implémentées dans Dedukti, permettant d’augmenter grandement sa fiabilité. Mots clés : théorie de la preuve, théorie des types, méthodes formelles Typechecking in the λΠ-Calculus Modulo: Theory and Practice Abstract: Automatic proof checking is about using a computer to check the validity of proofs of math- ematical statements. Since this verification is purely computational, it offers a high degree of confidence. Therefore, it is particularly useful for checking that a critical software, i.e., a software that when malfunc- tioning may result in death or serious injury to people, loss or severe damage to equipment or environmental harm, corresponds to its specification. Dedukti is such a proof-checker. It implements a type system, the
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Intersection Types and Runtime Errors in the Pi-Calculus

Intersection Types and Runtime Errors in the Pi-Calculus

When encoding the π -calculus in linear logic, π -calculus reduction is simulated by shallow proof net reduction (also known as surface reduction in linear λ-calculi [ Simpson 2005 ]). The concept of łshallownessž comes from the syntax of proof nets, which includes certain constructs called łboxesž, associated with the exponential modality !(−) and marking sub-proof nets that may be duplicated or erased. Shallow reduction never reduces inside a box. This is reminiscent of weak reduction in the λ-calculus, which does not reduce under an abstraction. Boxes are therefore a natural way of encoding the blocking behavior of prefixes in the π -calculus, i.e., the fact that π .P does not reduce even if P may reduce. Recall however that a box is associated with the modality !(−); so, if one kind of prefix (input or output) is associated with the presence of a box, then it will be associated with !(−), which implies that the dual kind of prefix will be associated with ?(−), which does not come with a box. Therefore, matching input/output duality with linear logical duality forces one prefix to be blocking and the other to be non-blocking; since it makes little sense for input to be non-blocking, this leads straight to the asynchronous π -calculus.
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Hide and New in the Pi-Calculus

Hide and New in the Pi-Calculus

Based on these considerations, in this paper we argue that the restriction operator of π-calculus does not adequately ensure confidentiality. To tackle this problem, we introduce an operator to program ex- plicitly secret communications, called hide. From a programming language point of view, the envisaged use of the operator is for declaring secret a medium used for local inter-process communication; exam- ples include pipelines, message queues and IPC mechanisms of microkernels. The operator is static: that is, we assume that the scope of hidden channels can not be extruded. The motivation is that all processes using a private channel shall be included in the scope of its hide declaration; processes outside the scope represent another location, and must not interfere with the protocol. Since the hide cannot extrude the scope of secret channels, we can use it to directly build specifications that preserves forward secrecy. In contrast, we regard the restriction operator of the π-calculus, new, as useful to create a new channel for message passing with scope extrusion, and which does not provide secrecy guarantees.
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The Lambda-calculus with multiplicities

The Lambda-calculus with multiplicities

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The Origins of lambda-calculus and term rewriting systems

The Origins of lambda-calculus and term rewriting systems

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Incrementality and effect simulation in the simply typed lambda calculus

Incrementality and effect simulation in the simply typed lambda calculus

The evaluation of a case-term considers the value of the term to be destructed. Then, the evaluation continues by comparing the value against the sequence tb 0  ¨ ¨ ¨  b m u, the comparison is denoted by ù. When comparing a term (a full applied constructor) with a branch, the semantic rules check whether or not the constructor identifier is the same as the constructor of the head- branch in the sequence. If they are the same, the evaluation proceed under the extended environment using the values of the value-constructor and taking the body-term of the associated function. If the identifiers are different, then the comparison continues with the rest of the sequence. We ensure that this exhaustive process succeeds since the static semantics takes the elimination expression containing all cases for the corresponding con- structors of the inductive type I. Therefore at some point both constructors, the one of the value and the one in head-position, will coincide making progress in the evaluation.
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Lambda-Calculus, Combinators and the Comprehension Scheme

Lambda-Calculus, Combinators and the Comprehension Scheme

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Asynchrony and the Pi-calculus

Asynchrony and the Pi-calculus

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Dynamic Compartments in the Imperative Pi Calculus

Dynamic Compartments in the Imperative Pi Calculus

Dynamic compartments may change their nesting structure dynamically, by applying operations for compartment creation, removal and merging. These op- erations may influence the speed of diverse reactions within compartments, in particular when compartment volumes change (global to local interactions). Vice versa, local reactions within a single compartment may effect global numeric attributes such as volume and surface (local to global interaction). Various lan- guages for modeling systems with dynamic compartments were proposed for sys- tems biology [18, 14, 21], but none of them can express physical, chemical, and compartimental aspects in a uniform manner, while providing efficient stochastic simulation. Spatial languages such as the Brane Calculi [2] or BioAmbients [18] fix a particular set of operators on compartments, and provide a special pur- pose solution for these operations. The π-calculus with polyadic synchronization
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A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure

A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure

::: ; u n ] is a list of terms. While the structure of the usual  -calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzen-style sequent calculus. To express the basis of the isomorphism, we consider intuitionistic logic with the implication as sole connective. However we do not consider Gentzen's calculus LJ, but a calculus LJT which leads to restrict the notion of cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this  -calculus, a substitution operator and a
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Non-idempotent typing operators, beyond the lambda-calculus

Non-idempotent typing operators, beyond the lambda-calculus

1.1. FUNCTIONS AND TERMINATION 19 The Decision Problem Hilbert had complemented his consistency program with the so-called Entscheidungsproblem (literally, the Decision Problem), which consisted in finding an algorithmic procedure that, given a first order formula F and an (effective) set of axioms, would output True if F is a syntactic consequence of the axioms and False if not, in the case such a procedure existed. Gödel’s theorems were a shock for many mathematicians, philosophers and logicians, and moreover, they were a strong indication that an algorithm of the Entscheidungsproblem did not exist as well, which had not been hitherto suspected. However, Gödel’s results did not straightforwardly give this negative answer, because their proof did not address the topic of computation, which was essential to understand what algorithmic procedures are and how they behave. Thus, the notion of computation, that had actually been overlooked by mathematicians and by logicians since the introduction of mathematics, came into light and caused intense reflection on its nature. Several alternative paradigms were proposed to provide a formal and comprehensive definition of computation. In his proof of the incompleteness theorems, Gödel had considered some obviously computable functions that are nowadays known as the primitive recursive functions. Integrating some remarks from Herbrand, he then defined the set of (partial) recursive functions, despite the fact he did not believe them to capture all possible computations (see [100], chapter 17). Church, who had introduced the λ-calculus in 1928, was convinced that a function was effectively computable iff it could be encoded by a λ-term, but many researchers, including Gödel, were skeptical. Finally, Turing defined his celebrated abstract machine [104] model, ever since known as the Turing machines. Turing explicitly conceived his machines by emulating (i.e. imitating in an abstract way) the human mind, seen as a device having a finite number of possible states and a reading/writing head interacting with an infinite tape, that is empty at the beginning of the execution (except for finitely many symbols). Very roughly, this captured the idea that (1) a human mind (or a cluster thereof) can handle only a finite number of data (i.e. what is already written on the tape) and this, in finitely many ways (captured by a finite transition function) (2) a human being writes/erases one letter after the other. Last, the assumption that the tape is infinite gives rise to the possibility to conduct a computation (or a reasoning) without limitation in space or time (just, the computation or the reasoning must stop at some point), which is what the notions of decidability and computability are about.
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Determinacy in a synchronous pi-calculus

Determinacy in a synchronous pi-calculus

In order to guarantee determinacy in the context of CCS rendez-vous communication, it seems quite natural to restrict the calculus so that interaction is point-to-point, i.e., it involves exactly one sender and one receiver. 1 In a synchronous framework, the introduction of signal based communication offers an opportunity to move from point-to-point to a more general multi-way interaction mechanism with multiple senders and/or receivers, while preserving determinacy. In particular, this is the approach taken in the Esterel and SL [8] models. The SL model can be regarded as a relaxation of the Esterel model where the reaction to the absence of a signal within an instant can only happen at the next instant. This design choice avoids some paradoxical situations and simplifies the implementation of the model. The SL model has gradually evolved into a general purpose programming language for concurrent applications and has been embedded in various programming environments such as C, Java, Scheme , and Caml (see [7, 22, 16]). For instance, the Reactive ML language [16] includes a large fragment of the Caml language plus primitives to generate signals and synchronise on them. We should also mention that related ideas have been developed by Saraswat et al. [21] in the area of constraint programming.
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Call-by-Value Lambda-calculus and LJQ

Call-by-Value Lambda-calculus and LJQ

19th April 2007 Abstract LJQ is a focused sequent calculus for intuitionistic logic, with a simple re- striction on the first premiss of the usual left introduction rule for implication. In a previous paper we discussed its history (going back to about 1950, or be- yond) and presented its basic theory and some applications; here we discuss in detail its relation to call-by-value reduction in lambda calculus, establishing a connection between LJQ and the CBV calculus λ C of Moggi. In particular,

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On the Representation of References in the pi-calculus

On the Representation of References in the pi-calculus

HAL Id: hal-02895654 https://hal.archives-ouvertes.fr/hal-02895654v2 Preprint submitted on 10 Jul 2020 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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On the Representation of References in the Pi-Calculus

On the Representation of References in the Pi-Calculus

HAL Id: hal-03053368 https://hal.archives-ouvertes.fr/hal-03053368 Submitted on 19 Dec 2020 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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ASMs and Operational Algorithmic Completeness of Lambda Calculus

ASMs and Operational Algorithmic Completeness of Lambda Calculus

1.2 Lambda Calculus and Operational Completeness In this paper we consider lambda calculus, a subject created by Church and Kleene in the 30’s, which enjoys a very rich mathematical theory. It may seem a priori strange to look for operational completeness with such a computation model so close to an assembly language (cf. Krivine’s papers since 1994, e.g., [22]). It turns out that, looking at reductions by groups (with an appropriate but constant length), and allowing one step reduction of primitive operations, lambda calculus simulates ASMs in a very tight way. Formally, our translation of ASMs in lambda calculus is as follows. Given an ASM, we prove that, for every integer K big enough (the least such K depending on the ASM), there exists a lambda term θ with the following property. Let a t
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On the confluence of lambda-calculus with conditional rewriting

On the confluence of lambda-calculus with conditional rewriting

assumed. However, it would be interesting to know if this restriction can be dropped. Problems arising from non left-linear rewriting are directly transposed to left-linear conditional rewriting. The semi-closure condition is sufficient to avoid this, and it seems to provide the counterpart of left-linearity for unconditional rewriting. However, two remarks have to be made about this restriction. First, it would be interesting to know if it is a necessary condition and besides, to characterize a class of non semi-closed systems that can be translated into equivalent semi-closed ones. Second, semi-closed terminating join systems behave like normal systems. But normal systems can be easily translated into equivalent non-conditional systems. Moreover such a translation preserves good properties such as left-linearity and non ambiguity. As many practical uses of rewriting rely on terminating systems, semi-closed join systems may be in practice essentially an intuitive way to design rewrite systems that can be then efficiently implemented by non-conditional rewriting.
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